Nucleation dynamics in two-dimensional cylindrical Ising models and chemotaxis

The aim of our work is to study the effect of geometry variation on nucleation times and to address its role in the context of eukaryotic chemotaxis (i.e., the process which allows cells to identify and follow a gradient of chemical attractant). As a first step in this direction we study the nucleation dynamics of the two-dimensional Ising model defined on a cylindrical lattice whose radius changes as a function of time. Geometry variation is obtained by changing the relative value of the couplings between spins in the compactified (vertical) direction with respect to the horizontal one. This allows us to keep the lattice size unchanged and study in a single simulation the values of the compactification radius which change in time. We show both with theoretical arguments and numerical simulations that squeezing the geometry allows the system to speed up nucleation times even in presence of a very small energy gap between the stable and the metastable states. We then address the implications of our analysis for directional chemotaxis. The initial steps of chemotaxis can be modeled as a nucleation process occurring on the cell membrane as a consequence of the external chemical gradient (which plays the role of energy gap between the stable and metastable phases). In nature most of the cells modify their geometry by extending quasi-one-dimensional protrusions (filopodia) so as to enhance their sensitivity to chemoattractant. Our results show that this geometry variation has indeed the effect of greatly decreasing the time scale of the nucleation process even in presence of very small amounts of chemoattractants.

Figure 1

(Color online) Site-dependent external magnetic field for several values of $ϵ$ with $c=1$. On the vertical axis is reported the site-dependent external magnetic field $h\left(x\right)$ while on the horizontal axis $x$ runs on the $L_x$ lattice side.

Figure 2

(Color online). $W=\frac{1}{2}\text{asinh}\left(\frac{k}{\text{sinh}\left(2K\right)}\right)$ for several values of $k$. On the $y$ and $x$ axes are reported $W$ and $K$, respectively. In the inset the points $\left(K;W\right)$ corresponding to ${\xi }_y=\frac{L_y}{2}$ are plotted: the symbol refers to the point (0.0113,2.53) for $k=1.8$, the # symbol refers to the point (0.0054,3.9) for $k=7$ and the X symbol refers to the point (0.0054,3.9) for $k=13.2$.

Figure 3

(Color online) Order parameters $\sigma$ and $M$ as functions of Monte Carlo time for three values of $k$, $ϵ=0.5$ and symmetrical couplings $K=W$. From the top to the bottom of the figure $\sigma$ and $M$ for $k=1.8$ in black, for $k=7$ in orange (gray) and for $k=13.2$ in yellow (light gray). The data for $k=7$ and $k=13.2$ oscillate around the same mean value.

Figure 4

(Color online) Order parameter $\sigma$ as function of Monte Carlo time for $k=13.2$ and several values of $ϵ$ for a simulation following protocol I $K=W=1$. From the top to the bottom of the figure, in orange (gray) data for $ϵ=0.005$, in black for $ϵ=0.5$, in red (dark gray) for $ϵ=0.05$, and in yellow (light gray) for $ϵ=0.5$ and symmetrical couplings.

Figure 5

(Color online) Order parameter $\sigma$ as function of Monte Carlo time for $k=13.2$ and several values of $ϵ$ for a simulation following protocol II $K=W=1$. From the top to the bottom of the figure, data for $ϵ=0.5$ in black, for $ϵ=0.05$ in red (dark gray), for $ϵ=0.005$ in orange (gray), and for $ϵ=0.5$ and symmetrical couplings in yellow (light gray).

Figure 6

(Color online) The figure shows several snapshots of a square lattice with side $L=100$ following protocol III [panel (a)] and protocol I [panel (b)]. Each snapshot is taken at a particular Monte Carlo step, indicated by the number of the corresponding iteration (iteration No. 1, iteration No. $30000$, and so on) below each square lattice. Blue (black) dots are up spins. In the panel (a), lattice with $K=W$ for the whole simulation, $k=13.2$ and $ϵ=0.5$. In the panel (b), coarsening dynamics in presence of asymmetrical couplings between $5\times {10}^3$ and $55\times {10}^3$ iterations (from the second to the fifth lattice), $k=13.2$ and $ϵ=0.5$.